Matrix of stoichiometric coefficients

Consider a system with N chemical components undergoing a set of M reactions. Obviously, N > M. Define the A x M matrix of stoichiometric coefficients as... 

Note that the matrix of stoichiometric coefficients devotes a row to each of the N components and a column to each of the M reactions. We require the reactions to be independent. A set of reactions is independent if no member of the set can be obtained by adding or subtracting multiples of the other members. A set will be independent if every reaction contains one species not present in the other reactions. The student of linear algebra will understand that the rank of v must equal M. 

This displays the convention, tacitly assumed later, that the positive direction of a step corresponds to the advancement from left to right of the stated chemical equation. The matrix of stoichiometric coefficients for these reactions is shown in Table II. The diagonalization of the matrix in Table II gives the matrix in Table III, from which the steady-state mechanism is S + 2s2 + 2s3 + 2s4. In Horiuti s terminology the stoichiometric numbers are 1 for Sj and 2 for s2, s3, and s4. 

Since H+ and e are always together, let us regard H+ + e as a single component and write it simply as H +. The matrix of stoichiometric coefficients is given in Table IV, the diagonalization of which gives the matrix in Table V. From this we conclude that the general steady-state mechanism is as follows ... 

By diagonalizing the matrix of stoichiometric coefficients, we obtain the matrix given in Table VII. The seventh row shows that there is a simple... 

The diagonalization of the matrix of stoichiometric coefficients is simplified in this case if the rows are not ordered as in steps (35). The result of diagonalizing is given in Table XI. Then, using the methods of Section IV,B, we find all the direct mechanisms of Table XII for the overall reaction... 

Diagonalization of the matrix of stoichiometric coefficients gives Table XVI, from which we read off the general steady-state mechanism (37) and its overall reaction (38), where p, a, and (j> are unrestricted ... 

Let the isomers be denoted by 1, 2C, and 2T. Then diagonalizing the matrix of stoichiometric coefficients gives matrix of Table XIX. From the matrix of Table XIX we obtain the general steady-state mechanism (40) with p and a unrestricted ... 

Rank of matrix of stoichiometric coefficients of intermediates only. Number of intermediates in chemical system. 

Here vT is the transposed matrix of stoichiometric numbers and Tint is the matrix of stoichiometric coefficients for intermediates. Elements of the latter are taken to be negative if substance is consumed in a given reaction step, positive if it is formed, and zero if substance is not involved in the reaction step. Multiplication of matrix vT (P-by-s) by matrix Tmt (s-by-/tot) gives the matrix vTrint whose size is (P-by-/tot) (s is the number of steps). 

Therein, x1, y1 represent the concentrations corresponding to the Nr reference equations and xn, yn the concentrations corresponding to the remaining N - Nr equations. Accordingly, the matrix of stoichiometric coefficients is split into two parts v1, v11. For the details, the reader is referred to Ref. [13]. 

The number of independent reactions R can be found simply as the rank of the matrix of stoichiometric coefficients %J with dimension Sx r such that R < r. Different methods can be applied, such as reduction to triangular matrix by Gaussian elimination for small-size matrices, or computer methods for larger problems. 

This is a homogeneous system of c linear equations in the s unknowns Xj. Let s be the rank of the matrix of stoichiometric coefficients. If the stoichiometric equations are independent, the X, s are all zero, and thus s = s. 

This is an homogeneous system of s linear equations in the c unknowns 7i 72, . 7c- The rank of the matrix of stoichiometric coefficients being equal to s > s, it is possible to choose arbitrarily c — s coefficients 7) and to calculate the others from the Cramer system of the remaining equations. Thus, there exist c —s sets of independent solutions for the coefficients jj, i.e. the number of invariants is equal to c —s. As for stoichiometries, any linear combination of invariants is an invariant. 

The input of kinetic, thermodynamic and operating numerical data into the computer is a problem of numerical file, which is easy to solve, as soon as the constituents and reactions have been identified and numbered and the matrix of stoichiometric coefficients has been determined. 

Thus, the main problems concern firstly, the input of the reaction mechanism into the computer (problem of chemical notation) and secondly, the processing of the reaction mechanism itself (problem of chemical compiler). Let us point out that the knowledge of tile matrix of stoichiometric coefficients allows us to compute the partial derivatives of the reaction rates with respect to the concentrations, i.e. a Jacobian matrix which has been shown to play a central role in the numerical computations. 

VRef = square matrix of stoichiometric coefficients for R reference components in R reactions XRef = column vector of mole fractions for R reference components in liquid phase... 

The array of coefficients on the right-hand side of the equations is manipulated as a matrix, which may be denoted as S, the matrix of stoichiometric coefficients. In the example,... 

S generalized Bource term, Section V parameter in Barkelew s criterion, Section VI (T or none none) s reduced stoichiometric matrix S matrix of stoichiometric coefficients... 

So matrix of stoichiometric coefficients for key components Sr matrix of Btoichiometric coefficients for key reactions T absolute temperature (T)... 

Now consider the case where there are R independent reactions. We discuss the meaning of independent shortly. We use the index J to identify reactions, and sums over the range of the J index, including the scalar product of two R-dimensional vectors, are indicated with, rather than <,>. Each of the reactions is identified by its own set of stoichiometric coefficients, so that we will have anNX R matrix of stoichiometric coefficients a,j [compactly indicated as a. In the present interpretation, a in Eq. (2) is an N X 1 matrix]. There are now R extents of reaction, which form an / -dimensional vector q. The obvious extension of Eq. (2) is... 

Solution First, we determine the number of independent reactions. We construct a matrix of stoichiometric coefficients for the given reactions ... 

We can construct the matrix of stoichiometric coefficients and reduce it to a diagonal form to determine the number of independent reactions. However, in this case, we have three reversible reactions, and, since each of the three forward reactions has a species that does not appear in the other two, we have three independent reactions and three dependent reactions. We select the three forward reactions as the set of independent reactions. Hence, the indices of the independent reactions are m = 1, 3, 5, and we describe the reactor operation in terms of their dimensionless extents, Zi, Z3, and Z5. The indices of the dependent reactions are = 2, 4, 6. Since this set of independent reactions consists of chemical reactions whose rate expressions are known, the heuristic rule on selecting independent reactions is satisfied. The stoichiometric coefficients of the selected three independent reactions are... 

Solution The purpose of this example is to illustrate the design formulations for chemical reactions where the dependency between dependent reactions and independent reactions is not due to reversible reactions. First, we determine the number of independent reactions and select a set of independent reactions. We construct a matrix of stoichiometric coefficients for the given reactions ... 

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